On Epsilon-Nets, Distance Oracles, and Metric Embeddings

TL;DR

This paper introduces new applications of a graph distance observation, including a near-linear size data structure for large distances and a metric embedding transformation focusing on preserving large distances.

Contribution

It presents two novel applications of an existing observation: a fast, compact data structure for large graph distances and a transformation for $\,ell_1$-embeddability emphasizing large distances.

Findings

Near-linear size data structure for large distances

Transformation of $\,ell_1$-embeddability results

Focus on preserving large distances

Abstract

We give two new applications of an observation from \cite{ADFGW11}. The first is an almost linear sized constant time data structure for reporting very large distances in undirected graphs. The second is a generic transformation of results about $\ell_1$-embeddability of metrics to a setting, where we are interested in preservation of large distances only.